We investigate characteristics of random split trees introduced by Devroye [SIAM J Comput 28, 409–432, 1998]; split trees include e.g., binary search trees, m-ary search trees, quadtrees, median of (2k + 1)-trees, simplex trees, tries and digital search trees. More precisely: We use renewal theory in the studies of split trees, and use this theory to prove several results about split trees. A s...